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Wednesday, June 26, 2013

Ever since the early 20th century when primitive analog computers were built to help compute solutions for naval gunnery fire control and increasing bomb accuracy, computing machinery has been used for weaponry. This trend continues to accelerate into the 21st century and has become an international competition.

Once upon a time

I had an early gift for mathematics and understanding three-dimensional form. When I was 16 or so, I helped my dad understand and then solve specific problems in spherical trigonometry. It eventually became clear to me that I was helping him verify circuitry specifically designed for suborbital mechanics: inertial guidance around the earth. Later I found out in those years he was working on the Poseidon SLBM for Lockheed, so, without completely understanding it, I was actually working on weaponized computation.

This is the period of my life where I learned about the geoid: the specific shape of the earth, largely an oblate ellipsoid. The exact shape depends upon gravitation, and thus mass concentrations (mascons). Lately the gravitational envelope of the moon caused by mascons has been an issue for the Lunar Orbiters.

At that point in history, rocket science was quite detailed and contained several specialized areas of knowledge. Many of which were helped by increasingly complex calculations. But there have been other fields that couldn't have advanced, where specific problems couldn't be solved, without the advances in computation. Ironically, some basic advances in computation we enjoy today owe these problems for their very existence. Consider this amazing article that details the first 25 years or so of the supercomputing initiatives at Lawrence Livermore National Laboratory.

Bombs

Throughout our computing history, computation has been harnessed to aid our defense by helping us create ever more powerful weapons. During the Manhattan Project at Los Alamos, Stanley Frankel and Eldred Nelson organized the T5 hand-computing group, a calculator farm populated with Marchant, Friden, and Monroe calculators and the wives of the physicists entering data on them. This group was arranged into an array to provide one of the first parallel computation designs, using Frankel's elegant breakdown of the computation into simpler, more robust calculations. Richard Feynman, a future Nobel prize winner, actually learned to fix the mechanical calculators so the computation could go on unabated by the huge time-sync of having to send them back to the factory for repair.

I was fortunate enough to be able to talk with Feynman when I was at Caltech, and we discussed group T-5, quantum theory, and how my old friend Derrick Lehmer was blacklisted for having a Russian wife. He told me that Stanley Frankel was also blacklisted. Also, I found 20-digit Friden calculators particularly useful for my computational purposes when I was a junior in High School.

The hunger for computation continued when Edward Teller began his work on the Super, a bomb organized around thermonuclear fusion. This lead John von Neumann, when he became aware of the ENIAC project, to suggest that the complex computations required to properly understand thermonuclear fusion could be carried out on one of the world's first electronic computers.

Codebreaking

In the history of warfare, codebreaking has proven itself to be of primary strategic importance. It turns out that this problem is perfectly suited to solution using computers.

One of the most important first steps in this area was taken at Bletchley Park in Britain during World War II. There, in 1939, Alan Turing constructed the Bombe. This was an early electromechanical computer and it was specifically designed to break the cipher and daily settings used in the German Enigma machine.

This effort required huge amounts of work and resulted in the discovery of several key strategic bits of information that turned the tide of the war against the Nazis.

The mathematical analysis of codes and encoded information is actually the science of decryption. The work on this is never-ending. At the National Security Agency's Multiprogram Research Facility in Oak Ridge, Tennessee, hundreds of scientists and mathematicians work to construct faster and faster computers for cryptanalytic analysis. And of course there are other special projects.

That seems like it would be an interesting place to work. Except there's no sign on the door. Well, this is to be expected since security is literally their middle name!

And the NSA's passion for modeling people has recently been highlighted by Edward Snowden's leaks of a slide set concerning the NSA's metadata-colecting priorities. And those slides could look so much better!

Passwords

In the modern day, hackers have become a huge problem for national and corporate security. This is partly because, recently, many advances in password cracking have occurred.

The first and most important advance was when RockYou.com was hacked with an SQL injection attack and 32 million (14.3 million unique) passwords were posted online. With a corpus like this, password crackers suddenly were able to substantially hone their playbooks to target the keyspaces that contain the most likely passwords.

A keyspace can be something like "a series of up to 8 digits" or "a word of up to seven characters in length followed by some digits" or even "a capitalized word from the dictionary with stylish letter substitutions". It was surprising how many of the RockYou password list could be compressed into keyspaces that restricted the search space considerably. And that made it possible to crack passwords much faster.

Popular fads like the stylish substitution of "i" by "1" or "e" by "3" were revealed to be exceptionally common.

Another advance in password cracking comes because passwords are usually not sent in plaintext form. Instead, a hashing function is used to obfuscate them. Perhaps they are only stored in hashed form. So, in 1980 a clever computer security professor named Martin Hellman published a technique that vastly sped up the process of password cracking. All you need to do is keep a table of the hash codes around for a keyspace. Then, when you get the hash code, you just look it up in the table.

But the advent of super-fast computers means that it is possible to compute billions of cryptographic hashes per second, allowing the password cracker to iterate through an entire keyspace in minutes to hours.

This is enabled by the original design of the hashing functions, like SHA, DES, and MD5, all commonly used hashing functions. They were all designed to be exceptionally efficient (and therefore quick) to compute.

So password crackers have written GPU-enabled parallel computation of the hashing functions. These run on exceptionally fast GPUs like the AMD Radeon series and the nVidia Tesla series.

To combat these, companies have started sending their passwords through thousands of iterations of the hashing function, which dramatically increases the time required to crack passwords. But really this only means that more computation is required to crack them.

The Internet

Many attacks on internet infrastructure and on targeted sites depend upon massively parallel capabilities. In particular, hackers often use Distributed Denial of Service (DDoS) attacks to bring down perceived opponents. Hackers often use an array of thousands of computers, called a botnet, to access a web site simultaneously, overloading the site's capabilities.

Distributed computing is an emerging technology that depends directly on the Internet. Various problems can be split into clean pieces and solved by independent computation. These include peaceful projects such as the spatial analysis of the shape of proteins (folding@home), the search for direct gravitational wave emissions from spinning neutron stars (Einstein@home), the analysis of radio telescope data for extraterrestrial signals (SETI@home), and the search for ever larger Mersenne prime numbers (GIMPS).

But not only have hackers been using distributed computing for attacks, they have also been using the capability for password cracking. Distributed computing is well suited to cryptanalysis also.

Exascale weapons

Recently it has been discussed that high-performance computing has become a strategic weapon. This is not surprising at all given how much computing gets devoted to the task of password cracking. Now the speculation is, with China's Tianhe-2 supercomputer, that weaponized computing is poised to move up to the exascale. The Tianhe-2 supercomputer is capable of 33.86 petaflops, less than a factor of 30 from the exascale. Most believe that exascale computing will arrive around 2018.

High-performance computing (HPC) has continually been used for weapons research. A high percentage of the most powerful supercomputers over the past decade are to be found at Livermore, Los Alamos, and Oak Ridge.

Whereas HPC has traditionally been aimed at floating-point operations (where real numbers are modeled and used for the bulk of the computation) the focus of password cracking is integer operations. For this reason, GPUs are typically preferred because modern general-purpose GPUs are capable of integer operations and they are massively parallel. The AMD 7990, for instance, has 4096 shaders. A shader is a scalar arithmetic unit that can be programmed to perform a variety of integer or floating-point operations. Because a GPU comes on a single card, this represents an incredibly dense ability to compute. The AMD 7990 achieves 7.78 teraflops but uses 135W of power.

So it's not out of the question to amass a system with thousands of GPUs to achieve exascale computing capability.

I feel it is ironic that China has built their fastest computer using Intel Xeon Phi processors. With 6 cores in each, the Xeon Phi packs about 1.2 teraflops of compute power per chip! And it is a lower power product than other Xeon processors, at about 4.25 gigaflops/watt. The AMD Radeon 7990, on the other hand, has been measured at 20.75 gigaflops/watt. This is because shaders are much scaled down from a full CPU.

What is the purpose?

Taking a step back, I think a few questions should be asked about computation in general. What should computation be used for? Why does it exist? Why did we invent it?

If you stand back and think about it, computation has only one purpose. This is to extend human capabilities; it allows us to do things we could not do before. It stands right next to other machines and artifices of mankind. Cars were developed to provide personal transportation, to allow us to go places quicker than we could go using our own two feet. Looms were invented so we could make cloth much faster and more efficiently than using a hand process, like knitting. Telescopes were invented so we could see farther than we could with our own two eyes.

Similarly, computation exists so we can extend the capabilities of our own brains. Working out a problem with pencil and paper can only go so far. When the problems get large, then we need help. We needed help when it came to cracking the Enigma cipher. We needed help when it came to computing the cross-section of Uranium. Computation was instantly weaponized as a product of necessity and the requirements of survival. But defense somehow crossed over into offensive capabilities.

With the Enigma, we were behind and trying to catch up. With the A-bomb, we were trying to get there before they did. Do our motivations always have to be about survival?

And where is it leading?

It's good that computation has come out from under the veil of weapons research. But the ramifications for society are huge. Since the mobile revolution, we solve problems that can occur to any of us in real life, and build an app for it. So computation continues to extend our capabilities in a way that fulfills some need. Computation has become commonplace and workaday.

When I see a kid learn to multiply by memorizing a table of products, I begin to wonder whether these capabilities are really needed, given the ubiquity of computation we can hold in our hands. Many things taught in school seem useless, like cursive writing. Why memorize historical dates when we can just look it up in Wikipedia? It's better to learn why something happened then when.

More and more, I feel that we should be teaching kids how to access and understand the knowledge that is always at their fingertips. And when so much of their lives is spent looking at an iPad, I feel that kids should be taught social interaction and be given more time to play, exercising their bodies.

It is because knowledge is so easy to access that teaching priorities must change. There should be more emphasis on the understanding of basic concepts and less emphasis on memorization. In the future, much of our memories and histories are going to be kept in the cloud.

Fundamentally, it becomes increasingly important to teach creativity. Because access to knowledge is not enough. We must also learn what to do with the knowledge and how to make advancements. The best advancements are made by standing on the shoulders of others. But without understanding how things interrelate, without basic reasoning skills, the access to knowledge is pointless.

Sunday, June 16, 2013

The last time I wrote about three-dimensional thinking, I discussed impossible figures. They are fun ways to challenge our brains to see things in a different way. But to me they signify more than just artwork.

Different Angles

Looking at objects from different angles helps us understand their spatial structure.

Looking at a given subject from different angles is a requirement for creativity. But eventually, in your mind, you realize that reality itself is malleable, and this is the domain of dreams. And dreaming is good for creativity because it helps us get out of the box of everyday experience and use our vision in a new way.

The key

Once I asked myself a question about impossible objects: what is the key to making one?

The key trick which is used in impossible figures is this: locally possible globally impossible. In the case of a Penrose triangle (also called a Reutersvärd triangle because Oscar Reutersvärd was the first to depict it) local corners and pieces of objects are entirely possible to construct but the way they are globally connected is spatially impossible.

I have constructed another impossible figure which is included above. This figure contains several global contradictions, yet remains locally plausible. However, there are two global levels of impossibility in this figure. Let's consider what they are.

First off, there are plenty of locally plausible geometries depicted in the figure. For instance, the M figure is a totally real and constructible object in the real world.

My original drawing didn't actually have M's at the three corners. It was a Penrose triangle. To make the figure compact, I added the M's on each of the three corners of the Penrose triangle. This doesn't make the figure any more possible though. It just adds a little salt and pepper to the mix; it helps confuse the eye a bit.

The next part shows the three strands connected to the three loops that wrap around the Penrose triangle.

There is really nothing about this strong figure that is impossible either. It can be totally constructed in real space.

Actually, it is a nice figure by itself, standing alone. You can see each block sliding by itself through the set of blocks.

And further, I this this figure would make a good logo. It feels like an impossible figure even though it's perfectly realizable. And it can be depicted from any angle because it is an honest three-dimensional construction. I have an idea to construct one out of lucite or another transparent material.

The next part of the figure is the loop. Each loop wraps around one of the sides of the Penrose triangle and creates an interlocking impossible figure, a concept I have shown examples of before in this blog. For instance, there is the impossible Valknut.

But this is the first level of impossibility. Such a loop is not really constructible without bending the top face. In this way, it is related to the unending staircase of M. C. Escher's Ascending and Descending.

The second level of impossibility is, of course, the Penrose triangle itself. When it comes to levels of impossibility and a clean depiction of impossibility, consider Reutersvärd. Pretty much all of Reutersvärd's art contains this illusion as a key. Though, I would encourage you to look at all of his work, because individual pieces can be both stunning and subtle simultaneously.

The next impossible figure is another modification of the Penrose triangle, showing what happens when the blocks intersect each other.

Any two blocks may certainly intersect each other, but to have all three intersect each other in this way is a clear impossibility.

It would probably have been more striking to make the triangular space in the center a bit larger.

Impossible objects take imagination out of the real world and into a world that maybe could be. Perhaps it's the world of flying cars, of paper that can hold any image and quickly change to any other, or of people whose thoughts are interconnected by quantum entanglement. In such a world, imagination can fly free.

Thursday, June 6, 2013

For those of you who have been reading posts and checking out my artwork for a while, I have a small present. I have posted the full-sized versions of much of my special artwork from this blog on Pinterest.

Finally, if you click on into them, you can see the details and get an idea of how many hours I spent on these pieces. Sometimes a bit of work was so complicated it took several days to complete. Which totally explains why my posts take so long!

Enjoy!

I have other boards on Pinterest, some with cool patterns, clouds, and collections of things. Just a hobby, and I thought I'd share a bit.

Sunday, June 2, 2013

I love patterns. This all originally stems from my observations of nature's patterns. A lot of the objects I draw (and develop in code mathematically) come directly from nature.

Strikingly, nature will often conspire to produce objects of great beauty, ones which we cannot match without tremendous effort. An example of this are the basalt formations. Created by volcanic upwelling, great pressure leading to crystallization, and fracturing during cooling, they are nature's brilliant tessellations, awe-inspiring extrusions, and mad ravings simultaneously.

They resemble three-dimensional bar graphs. Their fracture pattern, in two dimensions, is a natural Voronoi diagram. I first saw this pattern in nature while observing the way that soap bubbles join. Without fully understanding it, this observation introduced me to the mathematical laws of geometry when I was very young. Little did I know that I would never stop trying to duplicate it.

In this post, I show you how I duplicated this particular kind of nature. And I did it in my style, as you can see.

To create a drawing of a basalt formation, I actually used a rendered Voronoi diagram, which you see here, transformed it into a subtle perspective, establishing two vanishing points. Then I made three copies arranged as layers in a way that approximated placing them on three-dimensional transparent layers at various depths. This was so I could see the levels, and so the third vanishing point could be right.

Of course, I used Painter's Free Transform to do this!

I kept each layer a little bit transparent so I could get an intuitive feeling for which layer was on the top and which layer was on the bottom. This technique is called depth-cueing.

As you can see, it worked pretty well. I stopped at three layers because I didn't want the drawing project to get too complicated. But, of course, like all of my projects, it soon did!

Next, on a new layer, I drew lines on top of the the lines that I wanted to represent the three-dimensional surface of the basalt formation. This meant choosing a three-dimensional height for each cell. The base layer that extended to the outside of the drawing was the lowest height, of course, and a second and third layer was built on top of it.

This causes cells to raise out of the base layer and appear to become extruded.

When I consulted some real images of basalt formations as a guide, I found that they were quite imperfect and usually were cracked, damaged, or eroded in some way.

I really wanted my drawing to represent a perfect un-eroded result.

I used an extra transparent layer (behind the layer with the lines) and marked each cell with a three-dimensional height index so I could be sure which heights corresponded with each cells. This told me where to put the shading and also told me how to interpret the extrusion lines.

This layer was for informational purposes only. You see here the original small layer with crudely drawn lines. It's actually kind of hard to see the three-dimensional relative positions of the cells in some cases, which is another reason I labelled each cell with a height index.

Once I had designed it, I found that the drawing was way too small to shade the way I like to (using a woodcut technique) and so I resized the image and went over each of the lines by hand to make it crystal clear at the new resolution.

That only took a few days.

Why? After resizing the image, I found that each line was unusually soft. This meant that I had to go over the lines with a small brush, darkening and resolving the line. Then I had to go around it with white to create a clean edge. This is what really took the time!

Naturally I do lots of other things than just draw all the time, and so I had to use extra minutes here and there. I kept the Painter file on my laptop and brought my Wacom tablet with me in my bag.

I spent probably ten or twenty hours drawing this image.

Once the lines were perfect, the next step was shading. But of course it had to be in my style, and this also took quite a bit of time.

I used woodcut shading to create shadows and accessibility shading. This created a very nice look.

To do this, I drew parallel lines at a desired spacing, taking care to make them correspond in length and position to the shading and shadows that would result from a light coming from the left side.

I thickened the lines at their base, and made them a bit triangular. Then at the end, I used a small white brush to erode and sharpen the point and clean the sides of each shading line to get the right appearance.

The final step was coloring the tops and the sides, using a gel layer.

I colored each layer using a different shade of slightly bluish gray. The top layer got the lightest shade.

Here you can see a close-up of the final image, which was very high resolution indeed.

Even though I started out with a computer-generated fracturing pattern, I was able to retain a hand-wrought look to the final image. None of the lines are really computer-prefect